Unrestricted products of contractions in Banach spaces

Abstract

Let $X$ be a reflexive Banach space such that for any $x\ne 0$ the set $$\{x^{\ast }\in X^{\ast }:\text{∥x∗∥=1 and x∗(x)=∥x∥}\}$$ is compact. We prove that any unrestricted product of of a finite number of $(W)$ contractions on $X$ converges weakly.